A New Discussion on Banach-Type Fixed Point Result in Double-Composed Metric Spaces

Gwang-Myong Kim; Gum-Sik Kang; Chol Jin Kil; Jin-Sim Kim

doi:doi:10.11648/j.engmath.20250902.11

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A New Discussion on Banach-Type Fixed Point Result in Double-Composed Metric Spaces

Gwang-Myong Kim

, Gum-Sik Kang

, Chol Jin Kil

, Jin-Sim Kim^*

Published in Engineering Mathematics (Volume 9, Issue 2)

Received: 4 September 2025 Accepted: 30 September 2025 Published: 30 October 2025

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Abstract

One of the important research directions of fixed point theory is the generalization of metric spaces. An interesting generalization of metric space is the modification of triangular inequality. In the last few decades, many generalizations of metric space have been introduced in the field of fixed point theory by changing triangle inequality using multiplication of constants or functions. Recently, a new generalization of metric space with changing triangle inequality using the composition of two functions, namely, a double-composed metric space, has been introduced. A double-composed metric space is a new concept using the composition of functions, unlike the previous generalizations of metric space that modify the triangular inequality using the multiplication of functions. And, Banach-type fixed point result and Kanan-type fixed point result are established under certain assumptions in the setting of double-composed metric spaces. In this paper, we reconsider the Banach-type fixed point result in the setting of double-composed metric spaces under new and simple conditions. We have proved the fixed point theorem by using a new proof method and, consequently, we have demonstrated that Banach’s contractions in double-composed metric spaces have a unique fixed point under different assumptions from the previous one. We also present an example showing the validity of our fixed point result. Finally, we apply our fixed point result to show the existence of solution of Fredholm integral equations.

Published in	Engineering Mathematics (Volume 9, Issue 2)
DOI	10.11648/j.engmath.20250902.11
Page(s)	26-30
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Fixed Point, Banach Contraction, Double-Composed Metric Space, Fredholm Integral Equation

References

[1]	A. Bucur, About applications of the fixed point theory, Scientific Bulletin, 2017, XXII(1), 43.
[2]	S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fundamenta Mathematicae, 1922, 3(1), 133-181. https://doi.org/10.4064/fm-3-1-133-181
[3]	I. A. Bakhtin, The contraction mapping principle in almost metric spaces, Functional Analysis, 1989, 30, 26-37.
[4]	T. Kamran, M. Samreen, and Q. Ul Ain, A generalization of b-metric space and some fixed point theorems, Mathematics, 2017, 5(2), 19. https://doi.org/10.3390/math5020019
[5]	N. Mlaiki, H. Aydi, N. Souayah, and T. Abdeljawad, Controlled metric type spaces and the related contraction principle, Mathematics, 2018, 6(10), 194. https://doi.org/10.3390/math6100194
[6]	T. Abdeljawad, N. Mlaiki, H. Aydi, and N. Souayah, Double controlled metric type spaces and some fixed point results, Mathematics, 2018, 6(12), 320. https://doi.org/10.3390/math612320
[7]	I. Ayoob, N. Z. Chuan, and N. Mlaiki, Double-Composed Metric Spaces, Mathematics, 2023, 11(8), 1866. https://doi.org/10.3390/math11081866
[8]	C. J. Kil, C. S. Yu, and U. C. Han, Fixed point results for some rational type contractions in double-composed metric spaces and applications, Informatica, 2023, 34(12), 105-130.
[9]	F. M. Azmi, I. Ayoob, N. Mlaiki, Exploring Double Composed Partial Metric Spaces: A Novel Approach to Fixed Point Theorems, Int. J. Anal. Appl., 2024, 22, 192. https://doi.org/10.28924/2291-8639-22-2024-192
[10]	A. A. Hijab, L. K. Shaakir, S. Aljohani, N. Mlaiki, Double composed metric-like spaces via some fixed point theorems, AIMS Math., 2024, 9(10), 27205-27219. https://dx.doi.org/10.3934/math.20241322
[11]	C. J. Kil, K. Ho, W. Yang, U. Kim, Triple-Composed Metric Spaces and Related Fixed Point Results With Application, Journal of Function Spaces, 2024, Article ID 6466538, 9 pages. https://doi.org/10.1155/2024/6466538
[12]	C. J. Kil, B. Kim, S. Jon, H. Rim, Discussion on fixed points of nonlinear F-contractions in partially ordered double-composed metric-like spaces, Asian-European Journal of Mathematics, 2025, 18(11), 1-19. https://dx.doi.org/10.1142/S1793557125500469
[13]	A. A. Hijab, L. Shaakir, S. Aljohani, N. M. Mlaiki, Common Fixed Point Results in Double-Composed Cone Metric-Like Spaces for Some Generalized Rational Contractions with Applications, European Journal of Pure and Applied Mathematics, 2025, 18(2), 6029.
[14]	A. A. Hijab, L. K. Shaakir, S. Aljohani, N. Mlaiki, Fredholm integral equation in composed-cone metric spaces, Bound. Value Probl., 2024, 2024, 64. https://doi.org/10.1186/s13661-024-01876-w
[15]	C. J. Kil, B. Kim, Un-Ryong Rim, Discussion on Fixed Points for (α, F, Φ)-Contractions in Double-Composed Metric Spaces, Boletin de la Sociedad Mathematica Mexicana, 2025, 31(118), 1-20. https://doi.org/10.1007/s40590-025-00802-z

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APA Style

Kim, G., Kang, G., Kil, C. J., Kim, J. (2025). A New Discussion on Banach-Type Fixed Point Result in Double-Composed Metric Spaces. Engineering Mathematics, 9(2), 26-30. https://doi.org/10.11648/j.engmath.20250902.11

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ACS Style

Kim, G.; Kang, G.; Kil, C. J.; Kim, J. A New Discussion on Banach-Type Fixed Point Result in Double-Composed Metric Spaces. Eng. Math. 2025, 9(2), 26-30. doi: 10.11648/j.engmath.20250902.11

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AMA Style

Kim G, Kang G, Kil CJ, Kim J. A New Discussion on Banach-Type Fixed Point Result in Double-Composed Metric Spaces. Eng Math. 2025;9(2):26-30. doi: 10.11648/j.engmath.20250902.11

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@article{10.11648/j.engmath.20250902.11,
  author = {Gwang-Myong Kim and Gum-Sik Kang and Chol Jin Kil and Jin-Sim Kim},
  title = {A New Discussion on Banach-Type Fixed Point Result in Double-Composed Metric Spaces
},
  journal = {Engineering Mathematics},
  volume = {9},
  number = {2},
  pages = {26-30},
  doi = {10.11648/j.engmath.20250902.11},
  url = {https://doi.org/10.11648/j.engmath.20250902.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20250902.11},
  abstract = {One of the important research directions of fixed point theory is the generalization of metric spaces. An interesting generalization of metric space is the modification of triangular inequality. In the last few decades, many generalizations of metric space have been introduced in the field of fixed point theory by changing triangle inequality using multiplication of constants or functions. Recently, a new generalization of metric space with changing triangle inequality using the composition of two functions, namely, a double-composed metric space, has been introduced. A double-composed metric space is a new concept using the composition of functions, unlike the previous generalizations of metric space that modify the triangular inequality using the multiplication of functions. And, Banach-type fixed point result and Kanan-type fixed point result are established under certain assumptions in the setting of double-composed metric spaces. In this paper, we reconsider the Banach-type fixed point result in the setting of double-composed metric spaces under new and simple conditions. We have proved the fixed point theorem by using a new proof method and, consequently, we have demonstrated that Banach’s contractions in double-composed metric spaces have a unique fixed point under different assumptions from the previous one. We also present an example showing the validity of our fixed point result. Finally, we apply our fixed point result to show the existence of solution of Fredholm integral equations.
},
 year = {2025}
}

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TY  - JOUR
T1  - A New Discussion on Banach-Type Fixed Point Result in Double-Composed Metric Spaces

AU  - Gwang-Myong Kim
AU  - Gum-Sik Kang
AU  - Chol Jin Kil
AU  - Jin-Sim Kim
Y1  - 2025/10/30
PY  - 2025
N1  - https://doi.org/10.11648/j.engmath.20250902.11
DO  - 10.11648/j.engmath.20250902.11
T2  - Engineering Mathematics
JF  - Engineering Mathematics
JO  - Engineering Mathematics
SP  - 26
EP  - 30
PB  - Science Publishing Group
SN  - 2640-088X
UR  - https://doi.org/10.11648/j.engmath.20250902.11
AB  - One of the important research directions of fixed point theory is the generalization of metric spaces. An interesting generalization of metric space is the modification of triangular inequality. In the last few decades, many generalizations of metric space have been introduced in the field of fixed point theory by changing triangle inequality using multiplication of constants or functions. Recently, a new generalization of metric space with changing triangle inequality using the composition of two functions, namely, a double-composed metric space, has been introduced. A double-composed metric space is a new concept using the composition of functions, unlike the previous generalizations of metric space that modify the triangular inequality using the multiplication of functions. And, Banach-type fixed point result and Kanan-type fixed point result are established under certain assumptions in the setting of double-composed metric spaces. In this paper, we reconsider the Banach-type fixed point result in the setting of double-composed metric spaces under new and simple conditions. We have proved the fixed point theorem by using a new proof method and, consequently, we have demonstrated that Banach’s contractions in double-composed metric spaces have a unique fixed point under different assumptions from the previous one. We also present an example showing the validity of our fixed point result. Finally, we apply our fixed point result to show the existence of solution of Fredholm integral equations.

VL  - 9
IS  - 2
ER  -

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Author Information

Gwang-Myong Kim

Faculty of Applied Mathematics, Kim Chaek University of Technology, Pyongyang, Democratic People’s Republic of Korea

Contact Email

http://orcid.org/0009-0006-5810-0487
Gum-Sik Kang

Faculty of Applied Mathematics, Kim Chaek University of Technology, Pyongyang, Democratic People’s Republic of Korea

Contact Email

http://orcid.org/0009-0007-3525-822X
Chol Jin Kil

Faculty of Applied Mathematics, Kim Chaek University of Technology, Pyongyang, Democratic People’s Republic of Korea

Contact Email

http://orcid.org/0009-0008-9236-7139
Jin-Sim Kim

International Technology Cooperation Center, Kim Chaek University of Technology, Pyongyang, Democratic People’s Republic of Korea

Contact Email

http://orcid.org/0000-0002-7937-9185

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Plain Text BibTeX RIS

APA Style

Kim, G., Kang, G., Kil, C. J., Kim, J. (2025). A New Discussion on Banach-Type Fixed Point Result in Double-Composed Metric Spaces. Engineering Mathematics, 9(2), 26-30. https://doi.org/10.11648/j.engmath.20250902.11

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ACS Style

Kim, G.; Kang, G.; Kil, C. J.; Kim, J. A New Discussion on Banach-Type Fixed Point Result in Double-Composed Metric Spaces. Eng. Math. 2025, 9(2), 26-30. doi: 10.11648/j.engmath.20250902.11

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AMA Style

Kim G, Kang G, Kil CJ, Kim J. A New Discussion on Banach-Type Fixed Point Result in Double-Composed Metric Spaces. Eng Math. 2025;9(2):26-30. doi: 10.11648/j.engmath.20250902.11

Copy | Download

@article{10.11648/j.engmath.20250902.11,
  author = {Gwang-Myong Kim and Gum-Sik Kang and Chol Jin Kil and Jin-Sim Kim},
  title = {A New Discussion on Banach-Type Fixed Point Result in Double-Composed Metric Spaces
},
  journal = {Engineering Mathematics},
  volume = {9},
  number = {2},
  pages = {26-30},
  doi = {10.11648/j.engmath.20250902.11},
  url = {https://doi.org/10.11648/j.engmath.20250902.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20250902.11},
  abstract = {One of the important research directions of fixed point theory is the generalization of metric spaces. An interesting generalization of metric space is the modification of triangular inequality. In the last few decades, many generalizations of metric space have been introduced in the field of fixed point theory by changing triangle inequality using multiplication of constants or functions. Recently, a new generalization of metric space with changing triangle inequality using the composition of two functions, namely, a double-composed metric space, has been introduced. A double-composed metric space is a new concept using the composition of functions, unlike the previous generalizations of metric space that modify the triangular inequality using the multiplication of functions. And, Banach-type fixed point result and Kanan-type fixed point result are established under certain assumptions in the setting of double-composed metric spaces. In this paper, we reconsider the Banach-type fixed point result in the setting of double-composed metric spaces under new and simple conditions. We have proved the fixed point theorem by using a new proof method and, consequently, we have demonstrated that Banach’s contractions in double-composed metric spaces have a unique fixed point under different assumptions from the previous one. We also present an example showing the validity of our fixed point result. Finally, we apply our fixed point result to show the existence of solution of Fredholm integral equations.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - A New Discussion on Banach-Type Fixed Point Result in Double-Composed Metric Spaces

AU  - Gwang-Myong Kim
AU  - Gum-Sik Kang
AU  - Chol Jin Kil
AU  - Jin-Sim Kim
Y1  - 2025/10/30
PY  - 2025
N1  - https://doi.org/10.11648/j.engmath.20250902.11
DO  - 10.11648/j.engmath.20250902.11
T2  - Engineering Mathematics
JF  - Engineering Mathematics
JO  - Engineering Mathematics
SP  - 26
EP  - 30
PB  - Science Publishing Group
SN  - 2640-088X
UR  - https://doi.org/10.11648/j.engmath.20250902.11
AB  - One of the important research directions of fixed point theory is the generalization of metric spaces. An interesting generalization of metric space is the modification of triangular inequality. In the last few decades, many generalizations of metric space have been introduced in the field of fixed point theory by changing triangle inequality using multiplication of constants or functions. Recently, a new generalization of metric space with changing triangle inequality using the composition of two functions, namely, a double-composed metric space, has been introduced. A double-composed metric space is a new concept using the composition of functions, unlike the previous generalizations of metric space that modify the triangular inequality using the multiplication of functions. And, Banach-type fixed point result and Kanan-type fixed point result are established under certain assumptions in the setting of double-composed metric spaces. In this paper, we reconsider the Banach-type fixed point result in the setting of double-composed metric spaces under new and simple conditions. We have proved the fixed point theorem by using a new proof method and, consequently, we have demonstrated that Banach’s contractions in double-composed metric spaces have a unique fixed point under different assumptions from the previous one. We also present an example showing the validity of our fixed point result. Finally, we apply our fixed point result to show the existence of solution of Fredholm integral equations.

VL  - 9
IS  - 2
ER  -

Copy | Download